Electrically-tunable lenses and lens systems

ABSTRACT

An optical device ( 34, 66, 76 ) includes an electro-optical layer ( 48 ), having an effective local index of refraction at any given location within an active area of the electro-optical layer that is determined by a voltage waveform applied across the electro-optical layer at the location. Conductive electrodes ( 44, 64, 74, 82, 84 ) extend over opposing first and second sides of the electro-optical layer. The electrodes include an array of excitation electrodes, which extend along respective, mutually-parallel axes in a predefined direction across the first side of the electro-optical layer, and which includes at least first and second electrodes having different, respective widths in a transverse direction, perpendicular to the axes. Control circuitry ( 38 ) is coupled to apply respective control voltage waveforms to the excitation electrodes so as to generate a specified phase modulation profile in the electro-optical layer.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application claims the benefit of U.S. Provisional PatentApplication 61/952,226, filed Mar. 13, 2014; U.S. Provisional PatentApplication 61/969,190, filed Mar. 23, 2014; and U.S. Provisional PatentApplication 61/972,445, filed Mar. 31, 2014. This application is also acontinuation-in-part of PCT Patent Application PCT/IB2013/058989, filedSep. 30, 2013 (published as WO 2014/049577), which claims the benefit ofU.S. Provisional Patent Application 61/707,962, filed Sep. 30, 2012. Allof these related applications are incorporated herein by reference.

FIELD OF THE INVENTION

The present invention relates generally to optical devices, andparticularly to electrically-tunable lenses.

BACKGROUND

Tunable lenses are optical elements whose optical characteristics, suchas the focal length and/or the location of the optical axis, can beadjusted during use, typically under electronic control. Such lenses maybe used in a wide variety of applications. For example, U.S. Pat. No.7,475,985, whose disclosure is incorporated herein by reference,describes the use of an electro-active lens for the purpose of visioncorrection.

Electrically-tunable lenses typically contain a thin layer of a suitableelectro-optical material, i.e., a material whose local effective indexof refraction changes as a function of the voltage applied across thematerial. An electrode or array of electrodes is used to apply thedesired voltages in order to locally adjust the refractive index to thedesired value. Liquid crystals are the electro-optical material that ismost commonly used for this purpose (wherein the applied voltage rotatesthe molecules, which changes the axis of birefringence and thus changesthe effective refractive index), but other materials, such as polymergels, with similar electro-optical properties can alternatively be usedfor this purpose.

Some tunable lens designs use an electrode array to define a grid ofpixels in the liquid crystal, similar to the sort of pixel grid used inliquid-crystal displays. The refractive indices of the individual pixelsmay be electrically controlled to give a desired phase modulationprofile. (The term “phase modulation profile” is used in the presentdescription and in the claims to mean the distribution of the localphase shifts that are applied to light passing through the layer as theresult of the locally-variable effective refractive index over the areaof the electro-optical layer of the tunable lens.) Lenses using gridarrays of this sort are described, for example, in the above-mentionedU.S. Pat. No. 7,475,985.

The above-mentioned PCT International Publication WO 2014/049577describes optical devices that include an electro-optical layer, such asa liquid crystal, having an effective local index of refraction at anygiven location within the active area of the electro-optical layer thatis determined by a voltage waveform applied across the electro-opticallayer at the given location. An array of excitation electrodes,including parallel conductive stripes extending over the active area, isdisposed over one or both sides of the electro-optical layer. Controlcircuitry applies respective control voltage waveforms to the excitationelectrodes and is configured to concurrently modify the respectivecontrol voltage waveforms applied to excitation electrodes so as togenerate a specified phase modulation profile in the electro-opticallayer.

SUMMARY

Embodiments of the present invention that are described hereinbelowprovide improved electrically-tunable optical devices.

There is therefore provided, in accordance with an embodiment of theinvention, an optical device, which includes an electro-optical layer,having an effective local index of refraction at any given locationwithin an active area of the electro-optical layer that is determined bya voltage waveform applied across the electro-optical layer at thelocation. Conductive electrodes extend over opposing first and secondsides of the electro-optical layer. The electrodes include an array ofexcitation electrodes, which extend along respective, mutually-parallelaxes in a predefined direction across the first side of theelectro-optical layer, and which includes at least first and secondelectrodes having different, respective widths in a transversedirection, perpendicular to the axes. Control circuitry is coupled toapply respective control voltage waveforms to the excitation electrodesso as to generate a specified phase modulation profile in theelectro-optical layer.

In some embodiments, the respective widths of the electrodes differ fromone another with a standard variation that is at least 10% of a meanwidth of all the electrodes.

Additionally or alternatively, the respective widths of at least some ofthe electrodes vary along the respective axes of the electrodes.

In some embodiments, the array of excitation electrodes includes a firstarray of first excitation electrodes, extending in a first directionacross the first side of the electro-optical layer. The conductiveelectrodes includes a second array of second excitation electrodes,which extend in a second direction, perpendicular to the firstdirection, across the second side of the electro-optical layer, andwhich includes at least third and fourth electrodes having different,respective widths.

In other embodiments, the conductive electrodes include a commonelectrode, positioned over the active area on the second side of theelectro-optical layer. Some embodiments provides apparatus includingfirst and second optical devices of this sort, wherein the first andsecond optical devices are arranged in series, and wherein theexcitation electrodes in the second optical device are oriented in adirection orthogonal to the excitation electrodes in the first opticaldevice. In one embodiment, the first and second optical devices includerespective, first and second electro-optical layers that arepolarization-dependent and are arranged such that the first opticaldevice modulates light in a first polarization, while the second opticaldevice modulates the light in a second polarization, different from thefirst polarization, and the apparatus includes a polarization rotatorpositioned between the first and second optical devices so as to rotatethe light from the first polarization to the second polarization.

In some embodiments, the first and second electrodes have respectivefirst and second widths, such that the first width is at least twice thesecond width, and the control circuitry is configured to apply therespective control voltage waveforms so that the specified phasemodulation profile has an abrupt transition that occurs in a vicinity ofat least one of the second electrodes. In one such embodiment,generation of the specified phase modulation profile causes the deviceto function as a Fresnel lens. In a disclosed embodiment, the electrodesinclude parallel stripes of a transparent conductive material havinggaps between the stripes of a predefined gap width, and the second widthof the second electrodes is no greater than four times the gap width.Additionally or alternatively, the second width of the second electrodesis less than a layer thickness of the electro-optical layer.

In one embodiment, the phase modulation profile has multiple abrupttransitions that occur in respective vicinities of corresponding ones ofthe second electrodes, and the electro-optical layer is configured toprovide a range of phase modulation values that is proportional to arelation between a density of the second electrodes relative to aspacing between the abrupt transitions in the phase modulation function.

In some embodiments, the electro-optical layer includes a liquidcrystal.

There is also provided, in accordance with an embodiment of theinvention, an optical device, which includes an electro-optical layer asdescribed above, having opposing first and second sides and a layerthickness equal to a distance between the first and second sides.Conductive electrodes extend over the first and second sides of theelectro-optical layer. The electrodes include an array of excitationelectrodes including parallel stripes of a transparent conductivematerial having gaps between the stripes of a gap width that is lessthan the layer thickness of the electro-optical layer. Control circuitryis coupled to apply respective control voltage waveforms to theexcitation electrodes so as to generate a specified phase modulationprofile in the electro-optical layer.

In a disclosed embodiment, the gap width is less than half the layerthickness.

There is additionally provided, in accordance with an embodiment of theinvention, an optical device, which includes an electro-optical layer asdescribed above, and a buffer layer including a transparent dielectricmaterial having an interior surface adjacent to the first side of theelectro-optical layer and an exterior surface opposite the interiorsurface and a thickness of at least 0.2 μm between the interior andexterior surfaces. Conductive electrodes are disposed over the first andsecond sides of the electro-optical layer and include an array ofexcitation electrodes extending across the exterior surface of thebuffer layer, which separates the excitation electrodes from theelectro-optical layer. Control circuitry is coupled to apply respectivecontrol voltage waveforms to the excitation electrodes so as to generatea specified phase modulation profile in the electro-optical layer.

In a disclosed embodiment, the excitation electrodes include parallelstripes of a transparent conductive material having gaps between thestripes of a predefined gap width, and the buffer layer has a bufferlayer thickness that is more than one-fourth of the gap width.

There is further provided, in accordance with an embodiment of theinvention, an optical device, which includes an electro-optical layer asdescribed above. A first array of first excitation electrodes extendalong respective, mutually-parallel first axes in a first direction overthe active area on a first side of the electro-optical layer. A secondarray of second excitation electrodes extend along respective,mutually-parallel second axes in a second direction, orthogonal to thefirst direction, over the active area on a second side of theelectro-optical layer, opposite the first side. Control circuitry iscoupled to apply respective control voltage waveforms to the excitationelectrodes and is configured to concurrently modify the respectivecontrol voltage waveforms applied to both the first excitationelectrodes and the second excitation electrodes so as to generate aspecified phase modulation profile in the electro-optical layer.

In some embodiments, the phase modulation profile is defined as afunction that is separable into first and second component functions,which respectively vary along the first and second axes, and the controlvoltage waveforms applied to the first and second excitation electrodesare specified in accordance with the first and second componentfunctions, respectively. In one embodiment, the first and secondcomponent functions are defined in terms of a set of component waveformsthat are selected so as to correspond to different, respective phaseshifts in the electro-optical layer, such that the phase modulationprofile includes a sum of the respective phase shifts due to the firstand second component functions at each location within the active area.Typically, the component waveforms have different, respective dutycycles.

Additionally or alternatively, the component waveforms are selected sothat the sum of the respective phase shifts making up the phasemodulation profile is a modular sum with a modulus of 2nπ, wherein n isan integer and may have different, respective values for at least somedifferent pairs of the first and second component functions.

In some embodiments, the control voltage waveforms are selected so thatthe phase modulation profile contains abrupt phase transitions, and thedevice functions as a Fresnel lens. The control circuitry can beconfigured to apply the respective control voltage waveforms withopposite polarities to pairs of mutually-adjacent excitation electrodesin a vicinity of the abrupt phase transitions.

There is moreover provided, in accordance with an embodiment of theinvention, optical apparatus, which includes a static lens, including atransparent material having a curved exterior surface with a specifiedrefractive power and an interior surface containing at least first andsecond overlapping indentations. A dynamic lens is contained in thestatic lens and has a variable phase modulation profile, which modifiesthe refractive power of the static lens. The dynamic lens includes anelectro-optical layer, as described above, and first and secondtransparent substrates, which are disposed respectively on the first andsecond sides of the electro-optical layer and are sized and shaped tofit respectively into the first and second indentations in the staticlens, and which include electrodes configured to apply voltages acrossthe electro-optical layer. Control circuitry is coupled to apply thevoltages to the electrodes so as to generate the modulation profile inthe electro-optical layer.

In a disclosed embodiment, the control circuitry includes electricalconnections disposed at an edge of the first transparent substrate, andthe interior surface of the static lens contains a groove into which theelectrical connections fit.

There is furthermore provided, in accordance with an embodiment of theinvention, a method for producing an optical device. The method includesproviding an electro-optical layer, having an effective local index ofrefraction at any given location within an active area of theelectro-optical layer that is determined by a voltage waveform appliedacross the electro-optical layer at the location. Conductive electrodesare positioned over opposing first and second sides of theelectro-optical layer. The electrodes include an array of excitationelectrodes, which extend along respective, mutually-parallel axes in apredefined direction across the first side of the electro-optical layer,and which includes at least first and second electrodes havingdifferent, respective widths in a transverse direction, perpendicular tothe axes. Control circuitry is coupled to apply respective controlvoltage waveforms to the excitation electrodes so as to generate aspecified phase modulation profile in the electro-optical layer.

There is also provided, in accordance with an embodiment of theinvention, a method for producing an optical device, which includesproviding an electro-optical layer as described above, theelectro-optical layer having opposing first and second sides and a layerthickness equal to a distance between the first and second sides.Conductive electrodes are positioned over the first and second sides ofthe electro-optical layer. The electrodes include an array of excitationelectrodes including parallel stripes of a transparent conductivematerial having gaps between the stripes of a gap width that is lessthan the layer thickness of the electro-optical layer. Control circuitryis coupled to apply respective control voltage waveforms to theexcitation electrodes so as to generate a specified phase modulationprofile in the There is additionally provided, in accordance with anembodiment of the invention, a method for producing an optical device,which includes providing an electro-optical layer as described above,and positioning a buffer layer including a transparent dielectricmaterial having a thickness of at least 0.2 μm so that an interiorsurface of the buffer layer is adjacent to the first side of theelectro-optical layer. Conductive electrodes are positioned over thefirst and second sides of the electro-optical layer. The conductiveelectrodes include an array of excitation electrodes extending across anexterior surface of the buffer layer, which separates the excitationelectrodes from the electro-optical layer. Control circuitry is coupledto apply respective control voltage waveforms to the excitationelectrodes so as to generate a specified phase modulation profile in theelectro-optical layer.

There is further provided, in accordance with an embodiment of theinvention, a method for producing an optical device, which includesproviding an electro-optical layer as described above, and positioning afirst array of first excitation electrodes to extend along respective,mutually-parallel first axes in a first direction over the active areaon a first side of the electro-optical layer. A second array of secondexcitation electrodes is positioned to extend along respective,mutually-parallel second axes in a second direction, orthogonal to thefirst direction, over the active area on a second side of theelectro-optical layer, opposite the first side. Control circuitry iscoupled to apply respective control voltage waveforms to the excitationelectrodes and to concurrently modify the respective control voltagewaveforms applied to both the first excitation electrodes and the secondexcitation electrodes so as to generate a specified phase modulationprofile in the electro-optical layer.

There is moreover provided, in accordance with an embodiment of theinvention, a method for producing an optical device, which includesproviding a static lens, including a transparent material having acurved exterior surface with a specified refractive power and aninterior surface containing at least first and second overlappingindentations. A dynamic lens is embedded in the static lens, the dynamiclens having a variable phase modulation profile, which modifies therefractive power of the static lens. The dynamic lens includes anelectro-optical layer, as described above, and first and secondtransparent substrates, which are disposed respectively on the first andsecond sides of the electro-optical layer and are sized and shaped tofit respectively into the first and second indentations in the staticlens, and which include electrodes configured to apply voltages acrossthe electro-optical layer. Control circuitry is coupled to apply thevoltages to the electrodes so as to generate the modulation profile inthe electro-optical layer.

There is furthermore provided, in accordance with an embodiment of theinvention, an optical device, including an electro-optical layer asdescribed above, and conductive electrodes extending over opposing firstand second sides of the electro-optical layer. The electrodes include anarray of excitation electrodes, which extend along respective,mutually-parallel axes in a predefined direction across the first sideof the electro-optical layer, while respective center points of theelectrodes are displaced transversely by an amount that varies along therespective axes of the electrodes. Control circuitry is coupled to applyrespective control voltage waveforms to the excitation electrodes so asto generate a specified phase modulation profile in the electro-opticallayer.

In one embodiment, the array of the electrodes on the first side of theelectro-optical layer includes a first array of first excitationelectrodes, extending in a first direction across the first side of theelectro-optical layer, and the conductive electrodes include a secondarray of second excitation electrodes, which extend in a seconddirection, perpendicular to the first direction, across the second sideof the electro-optical layer, while respective center points of thesecond excitation electrodes are displaced transversely along therespective axes thereof.

There is also provided, in accordance with an embodiment of theinvention, an optical device, including an electro-optical layer asdescribed above, and conductive electrodes extending over the first andsecond sides of the electro-optical layer. The electrodes including anarray of excitation electrodes including parallel stripes of atransparent conductive material. Control circuitry is coupled to applyrespective control voltage waveforms to the excitation electrodes so asto generate in the electro-optical layer a specified phase modulationprofile containing abrupt phase transitions, while applying therespective control voltage waveforms with opposite polarities to pairsof mutually-adjacent excitation electrodes in a vicinity of the abruptphase transitions.

In a disclosed embodiment, the control voltage waveforms are selected soas to cause the device to function as a Fresnel lens.

There is additionally provided, in accordance with an embodiment of theinvention, a method for producing an optical device. The method includesproviding an electro-optical layer, as described above, and positioningconductive electrodes extending over opposing first and second sides ofthe electro-optical layer. The electrodes include an array of excitationelectrodes, which extend along respective, mutually-parallel axes in apredefined direction across the first side of the electro-optical layer,while respective center points of the electrodes are displacedtransversely by an amount that varies along the respective axes of theelectrodes. Control circuitry is coupled to apply respective controlvoltage waveforms to the excitation electrodes so as to generate aspecified phase modulation profile in the electro-optical layer.

There is further provided, in accordance with an embodiment of theinvention, a method for producing an optical device, which includesproviding an electro-optical layer, as described above, and positioningelectrodes over opposing first and second sides of the electro-opticallayer. The electrodes include an array of excitation electrodesincluding parallel stripes of a transparent conductive material. Controlcircuitry is coupled to apply respective control voltage waveforms tothe excitation electrodes so as to generate in the electro-optical layera specified phase modulation profile containing abrupt phasetransitions, while applying the respective control voltage waveformswith opposite polarities to pairs of mutually-adjacent excitationelectrodes in a vicinity of the abrupt phase transitions.

There is moreover provided, in accordance with an embodiment of theinvention, optical apparatus, which includes first and second opticaldevices, which have respective first and second polarization axes andfirst and second cylinder axes and are arranged in series such that thefirst and second polarization axes are mutually non-parallel and thefirst and second cylinder axes are mutually non-parallel. Each of theoptical devices includes a polarization-dependent electro-optical layer,having an effective local index of refraction, for light that ispolarized along the respective polarization axis and is incident at anygiven location within an active area of the electro-optical layer, thatis determined by a voltage applied across the electro-optical layer atthe location. Conductive electrodes extend over opposing first andsecond sides of the electro-optical layer. The electrodes include anarray of excitation electrodes, which is configured to apply respectivevoltages across the excitation electrodes so as to generate in theelectro-optical layer a cylindrical phase modulation profile orientedalong the respective cylinder axis. A polarization rotator is positionedbetween the first and second optical devices so as to rotate apolarization of light that has passed through the first optical deviceand is parallel to the first polarization axis, in order to align thepolarization of the light with the second polarization axis.

In some embodiments, the array of excitation electrodes includes anarray of parallel stripes of a transparent, conductive materialextending across the first side of the electro-optical layer in adirection parallel to the respective cylinder axis, and the conductiveelectrodes include a common electrode, positioned over the active areaon the second side of the electro-optical layer. Typically, the firstand second polarization axes are mutually perpendicular and the firstand second cylinder axes are mutually perpendicular.

The present invention will be more fully understood from the followingdetailed description of the embodiments thereof, taken together with thedrawings in which:

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a schematic side view of an optical system, in accordance withan embodiment of the present invention;

FIG. 2A is a schematic sectional view of an electrically-tunable lens,in accordance with an embodiment of the present invention;

FIG. 2B is a schematic frontal view of electrodes used in the device ofFIG. 2A;

FIG. 3A is a schematic frontal view of a compound lens containing staticand electrically-tunable components, in accordance with an embodiment ofthe present invention;

FIG. 3B is a schematic sectional view of the static component of thecompound lens of FIG. 3A, taken along the line III-III that is shown inFIG. 3A;

FIG. 3C is a schematic sectional view of the electrically-tunablecomponent of the compound lens of FIG. 3A, taken along the line III-IIIthat is shown in FIG. 3A;

FIGS. 4A and 4B are schematic frontal views of electrodes formed onopposing sides of an electrically-tunable lens, in accordance with anembodiment of the present invention;

FIG. 4C is a schematic frontal view of the device of FIGS. 4A and 4B,showing a superposition of the electrodes on the opposing sides of thedevice, in accordance with an embodiment of the present invention;

FIGS. 5A-5D are plots that schematically show the point spread functionsof retinal images formed using the device of FIGS. 4A-4C for differentlevels of pixel size variation, in accordance with an embodiment of thepresent invention;

FIGS. 6A and 6B are schematic frontal views of electrodes formed onopposing sides of an electrically-tunable lens, in accordance withanother embodiment of the present invention;

FIG. 6C is a schematic frontal view of the device of FIGS. 6A and 6B,showing a superposition of the electrodes on the opposing sides of thedevice, in accordance with an embodiment of the present invention;

FIGS. 7A and 7B are schematic frontal views of electrodes formed onopposing sides of an electrically-tunable lens, in accordance with analternative embodiment of the present invention;

FIG. 7C is a schematic frontal view of the device of FIGS. 7A and 7B,showing a superposition of the electrodes on the opposing sides of thedevice, in accordance with an embodiment of the present invention;

FIG. 8 is a schematic frontal view of electrodes in anelectrically-tunable lens, in accordance with an embodiment of thepresent invention;

FIG. 9 is a schematic frontal view of electrodes in anelectrically-tunable lens, in accordance with another embodiment of thepresent invention;

FIG. 10 is a plot that schematically shows variation in phase modulationbetween pixels in the device of FIG. 8 and in a device having pixels ofuniform width, in accordance with an embodiment of the presentinvention;

FIG. 11 is a plot that schematically shows variations in phasemodulation between adjacent electrodes of an electrically-tunable lensfor different gap widths between the electrodes, in accordance with anembodiment of the present invention;

FIG. 12 is a schematic sectional view of an electrically-tunable lenscomprising a buffer layer between the electrodes and a liquid crystal inthe lens, in accordance with an alternative embodiment of the presentinvention;

FIG. 13 is a plot that schematically shows variation in phase modulationbetween pixels in the device of FIG. 12 and in a device without a bufferlayer, in accordance with an embodiment of the present invention;

FIGS. 14A and 14B are schematic frontal views of electrodes formed onopposing sides of an electrically-tunable lens, in accordance with anembodiment of the present invention;

FIG. 14C is a schematic frontal view of the device of FIGS. 14A and 14B,showing a superposition of the electrodes on the opposing sides of thedevice, in accordance with an embodiment of the present invention;

FIG. 15 is a schematic representation of a phase modulation functiongenerated by the device of FIGS. 14A-14C when driven in accordance withan embodiment of the present invention;

FIG. 16 is a plot that schematically illustrates a phase modulationcurve of the device of FIGS. 14A-14C as a function of the voltageapplied across the electrodes, in accordance with an embodiment of thepresent invention;

FIGS. 17A-17D are plots that schematically illustrate voltage waveformsapplied to the electrodes of FIG. 14A, in accordance with an embodimentof the present invention;

FIGS. 18A-18D are plots that schematically illustrate voltage waveformsapplied to the electrodes of FIG. 14B, in accordance with an embodimentof the present invention;

FIGS. 19A-19G are plots that schematically illustrate voltage waveformsgenerated across the liquid crystal in an electrically-tunable lens as aresult of applying different combinations of the waveforms of FIGS.17A-17D and FIGS. 18A-18D to the electrodes; and

FIG. 20 is a schematic side view of an optical system, in accordancewith another embodiment of the invention.

DETAILED DESCRIPTION OF EMBODIMENTS Overview

Electrically-tunable lenses using an electro-optical material with apixel grid can, in principle, generate any desired phase modulationprofile, within the limits of the achievable range of the localrefractive index and the pitch of the pixels. Realization of practicallenses for many applications, however, such as for ophthalmic use,requires a large addressable grid of very small pixels, for example, anarray of at least 400×400 pixels with a pitch of 50 μm or less.

In liquid-crystal display (LCD) panels, the pixels are typicallyarranged in a matrix of N rows and M columns. Each of the N*M pixels canreceive a set of possible values (gray levels), independent of all otherpixels. Different pixel values are obtained by altering the localvoltages applied to the liquid crystal (LC) layer. Typically the voltageis time-varying and alternating in sign (AC), in some cases at a ratefaster than the LC response time, and the LC responds to the effectiveaverage applied voltage as long as the average voltage is above acertain threshold.

Embodiments of the present invention that are described herein providenovel electrically-tunable optical devices that are able to achieveenhanced optical performance relative to devices that are known in theart. The disclosed devices may be configured to operate as cylindricallenses (with focusing along one axis, using an array of stripeelectrodes) or to emulate spherical lenses, with dual-axis focusing.Both the focal power and the location of the optical axis, i.e., theeffective central point or axis of the lens defined in this manner, canbe changed freely and rapidly by application of appropriate controlvoltages. The disclosed embodiments build on the principles set forth inthe above-mentioned PCT/IB2013/058989, while adding novel and improvedfeatures. Devices constructed in accordance with the present embodimentsare advantageous particularly in ophthalmic lens systems but mayalternatively be used in other applications.

Generally speaking, the disclosed devices (which are referred to hereinas electrically-tunable lenses) can be configured to apply any phasemodulation profile that is separable. A two-dimensional phase modulationprofile e^(iø(x,y)) is separable if it can be decomposed into a productof two one-dimensional functions, e^(iø(x,y))=e^(iø) ^(x) ^((x))·e^(iø)^(y) ^((y)). In other words, the disclosed devices are able to apply anyphase modulation profile that is defined as a function that is separableinto two component functions that vary along respective,mutually-orthogonal axes, and the phase modulation profile will thencomprise a sum of the respective phase shifts due to the first andsecond component functions. As phase is a cyclical function, with period2π, the term “sum” should be understood in this context as includingmodular summation, with modulo 2nπ, wherein n is an integer. In someembodiments, n may have different, respective values for at least somedifferent pairs of the first and second component functions.

In some of the disclosed embodiments, an optical device comprises anelectro-optical layer, meaning, as explained above, that the localeffective index of refraction at any given location within the activearea of the layer is determined by the voltage applied across the layerin the vicinity of the location. Typically, the electro-optical layercomprises a liquid crystal (LC), possibly a polarization-independentliquid crystal layer (such as a cholesteric LC layer), although othertypes of electro-optical materials may alternatively be used. A commonelectrode is positioned over the active area on one side of theelectro-optical layer. An array of excitation electrodes, made from aconductive material and having mutually-parallel axes, extends over theactive area on the opposite side of the electro-optical layer.

To drive and modify the phase modulation profile of the electro-opticallayer, control circuitry applies respective control voltages to theexcitation electrodes. Typically, each excitation electrode isindividually connected to and controlled by the control circuitry, sothat the voltage waveforms applied to several or even all of theexcitation electrodes can be modified concurrently. This configurationprovides an optical element of arbitrary, tunable one-dimensionalprofile (such as a cylindrical lens), with a resolution limited only bythe distance between the electrodes and the thickness of theelectro-optical layer. The phase modulation properties of the device canbe modified at a rate limited only by the speed of the control circuitryand the response time of the electro-optical layer.

In some embodiments, two devices of this type are superimposed at rightangles, with the excitation electrodes in one device oriented in adirection orthogonal to those in the other device, in order to provideapparatus capable of emulating a spherical lens under the paraxialapproximation.

In other embodiments, an optical device comprises an electro-opticallayer with first and second arrays of excitation electrodes on oppositesides of the layer, with the second array oriented in a directionorthogonal to the first array. Control circuitry applies respectivecontrol voltage waveforms to the excitation electrodes in both arraysand is capable of modifying the control voltages applied to multipleexcitation electrodes (and possibly all of the excitation electrodes) onboth of the sides of the electro-optical layer. The control circuitrymay concurrently modify the respective control voltage waveforms appliedto the excitation electrodes in both the first and second arrays so asto generate a specified phase modulation profile in the electro-opticallayer.

In these latter embodiments, no time-division multiplexing scheme isrequired, and both row and column voltage waveforms are data-dependent.The voltage waveforms are chosen to create a linear phase response inthe electro-optical material. Formally stated, the term “linear phaseresponse,” used in connection with voltage waveforms applied to theelectro-optical layer, means that when a set of voltages waveforms{V_(X,i)(t)}_(i=1) ^(N) is applied to a first set of electrodes,corresponding to a first predefined set of phase values {Ø_(X,i)}_(i=1)^(N), and a set of voltage waveforms {V_(Y,j)(t)}_(j=1) ^(M) is appliedto a second set of electrodes, positioned orthogonally to the first setof electrodes, corresponding to a second predefined set of phase values{Ø_(Y,j)}_(j=1) ^(M), then for each i=1 . . . N and j=1 . . . M, thephase modulation profile T_(LC){V(t)} for light passing through theelectro-optical layer when the voltage waveform V(t) is applied to itwill be T_(LC){V_(X,i)(t)−V_(Y,j)(y)}=e^(i(ø) ^(X,i) ^(+ø) ^(Y,j) ⁾.

Thus, in distinction to LCD panels that are known in the art, both theX- and Y-axis electrodes are driven with data-dependent voltagewaveforms, and all electrodes can be driven concurrently andindependently. The term “concurrently,” as used in this context, meansthat the driving waveforms are applied to multiple pixels, in differentrows and columns of the array defined by the electrodes, at the sametime, without time-division multiplexing. The term “independently” meansthat a different, data-dependent waveform may be applied to eachelectrode, along both X- and Y-axes. The control circuitry may apply therespective voltages to different ones of the electrodes at differentamplitudes and/or with temporal waveforms, typically having differentduty cycles. (The term “duty cycle,” as used in the present descriptionand in the claims, refers to the fraction of the time during each periodof a given waveform in which the voltage of the waveform is non-zero.)

Some of the embodiments that are described herein are directed atachieving smooth, continuous phase modulation, by reducing sharp phasetransitions and diffraction effects that can occur due to the pixelatedstructure imposed by the electrodes in the device. In addition oralternatively, some of the present embodiments are directed to avoidingghost images caused by light scattering from periodic structures, suchas the pixels of an electronically-tunable lens, by avoiding the use ofstructures with distinct periods. For example, instead of the uniformstripe electrodes used in devices that are known in the art, in someembodiments of the present invention, the excitation electrodes havedifferent, respective widths (when measured in the transverse direction,perpendicular to the long axes of the electrodes). Different electrodesmay have different widths or, in some cases, the widths of at least someof the electrodes vary along the respective axes of the electrodesthemselves. Additionally or alternatively, when the phase modulationprofile of the device is meant to include abrupt transitions, as in aFresnel lens profile, these transitions may be made to occur atelectrodes with narrow widths, relative to the other electrodes, inorder to give sharper, more precise transitions.

Abrupt phase transitions also arise at undesired locations in pixelatedlens devices due to the gaps between adjacent electrodes. In someembodiments of the present invention, this problem is addressed bynarrowing the gap width, typically to less than the layer thickness ofthe electro-optical layer, and possibly to less than half the layerthickness. Additionally or alternatively, a buffer layer, comprising atransparent dielectric material, is interposed between the excitationelectrodes and the electro-optical layer and thus smooths the phasetransitions in the areas of the electro-optical layer that are adjacentto the gaps between the electrodes.

System Description

FIG. 1 is schematic side view of an optical system 20, in accordancewith an embodiment of the present invention. In the pictured embodiment,system 20 is configured to function as an ophthalmic lens, whichprovides dynamic correction for the vision of an eye 22 of a user. Thisembodiment is just one non-limiting example, however, of possibleapplications of the principles of the present invention.

System 20 comprises “static” lenses 24, 26 and 28, with a “dynamic” lens34 embedded in lens 26. Although three static lenses are shown here forthe sake of completeness, in many applications only one or two staticlenses are required, and it will often be sufficient to use a singlestatic lens, such as lens 26, in which the dynamic lens is embedded.Dynamic lens 34 comprises one or more electrically-tunable opticaldevices, which may be of any of the suitable forms that are describedherein. Lenses 24, 26 and 28 are “static” in the sense that theirrefractive powers are fixed. Lens 34 is dynamic in that it has avariable phase modulation profile, which modifies the refractive powerof optical system 20. Lenses 24, 26 and 28 provide the baselinerefractive power of system 20, which is dynamically adjusted byoperation of lens 34.

A control unit (not shown) controls dynamic lens 34 so as to tune itsoptical power and alignment. For example, the optical power may beincreased or decreased to accommodate the distance at which eye 22 isattempting to focus. Lens 34 may be set to emulate a spherical lens,possibly with the addition of aspheric components. Additionally oralternatively, lens 34 may function as an astigmatic lens.

As another example, which is illustrated in FIG. 1, the optical centerline of dynamic lens 34 may be shifted transversely, so that the opticalaxis of system 20 shifts from a baseline axis 30 to a deviated axis 32.This sort of axis shift can be applied, possibly in conjunction withtracking of the eye, to dynamically align the optical axis of the systemwith the user's gaze angle.

More generally speaking, lens 34 can be controlled, by application ofappropriate control voltages, to implement substantially any desiredphase profile that is separable into horizontal and vertical components,as long as the range of phase shifts in the profile is achievable by therange of refractive index variation and the thickness of theelectro-optical layer (or layers) in dynamic lens 34. To reduce therequired range of phase shifts, the control voltages may be chosen sothat lens 34 operates as a Fresnel lens.

FIGS. 2A and 2B schematically show details of electrically-tunabledynamic lens 34, in accordance with an embodiment of the presentinvention. FIG. 2A is a sectional view, while FIG. 2B is a frontal viewof excitation electrodes 44 used in the lens 34. In this example,excitation electrodes 44 are formed on only one side of anelectro-optical layer 48 in lens 34, with a common electrode 46 on theother side, so that the pictured components of lens 34 will function asa dynamic cylindrical lens. In such a case, lens 34 will typicallycomprise two such cylindrical lenses, with their axes oriented at rightangles to one another in order to emulate a two-dimensional lens.Alternatively, lens 34 may comprise two arrays of excitation electrodeson opposing sides the electro-optical layer, as is shown and describedin greater detail hereinbelow.

A pair of cylindrical lenses in series may be arranged to emulate atwo-dimensional lens in a number of different ways: When theelectro-optical layer is polarization-dependent (i.e., modulates lightonly in a certain polarization), the two cylindrical lenses may beconfigured and mounted so that their respective polarization axes aremutually parallel, even though the optical cylinder axes are orthogonal.Alternatively, when the first and second cylindrical lenses havenon-parallel axes of polarization (for example, with the polarizationaxes parallel to the respective cylinder axes), an additional opticalelement is positioned between the two cylindrical lenses in order torotate the light polarization from the first to the second polarization.This sort of arrangement is shown below in FIG. 20. Of course, if theelectro-optical layer is polarization-independent, then no specialmeasures are required in this regard.

Electro-optical layer 48, such as a liquid-crystal (LC) layer, istypically contained by suitable encapsulation, as is known in the art.The encapsulation may include sealing of the sides of the active area,to prevent leakage of the active material. Additionally oralternatively, the encapsulation may include alignment layers on the topand bottom substrates to ensure correct alignment of the molecules ofthe liquid crystal in the active area. For example, the alignment layercan comprise a thin layer of polyimide with rubbing defining themolecule alignment axis.

Layer 48 has a local effective index of refraction at any given locationwithin its active area (for example, within the area of layer 48 thatactually contains the liquid crystal) that is determined by the voltageapplied across the layer at that location. The liquid crystal in layer48 may be birefringent, in which case lens 34 or system 20 may comprisea polarizer, as is known in the art (omitted from the figures forsimplicity), in order to select the polarization of the light that is tobe passed and refracted by layer 48. Alternatively, to avoid the needfor a polarizer, two such lenses can be concatenated with perpendicularaxes of birefringence, so that each operates on a different, orthogonalpolarization; or a polarization-independent liquid crystal layer, suchas a layer of cholesteric liquid crystal material, may be used.

Transparent substrates 40 and 42, such as glass blanks, are positionedon opposing sides of layer 48, and electrodes 44 and 46 are disposed onthe substrates. The electrodes typically comprise a transparent,conductive material, such as indium tin oxide (ITO), as is known in theart. Alternatively, non-transparent electrodes may be used, as long asthey are thin enough so that they do not cause disturbing opticaleffects. Common electrode 46 on substrate 42 is positioned over theactive area of layer 48 on one side. An array of excitation electrodes44, comprising stripes of the transparent conductive material onsubstrate 40 with mutually-parallel axes 49, extends over the activearea on the opposite side of layer 48. (Axes 49 of the electrodes runalong the long dimension of the electrodes, and “parallel” in thiscontext may include, as well, electrodes that deviate in angle byseveral degrees.)

The electrode patterns shown in these and the other figures may beformed, for example, by lithography on substrates 40 and 42, after whichthe substrates are glued together at a predefined distance, typically afew microns, by using glues or etched spacers as are known in the art.Layer 48 is then inserted and sealed in the gap between the substrates.Although for the sake of visual clarity, only a few electrodes 44 areshown in FIG. 2B, in practice, for good optical quality, lens 34 willtypically comprise at least 100 stripe electrodes for excitation, andpossibly even 400 or more.

Control circuitry 38 is coupled to apply respective control voltages toexcitation electrodes 44, relative to the common voltage level ofelectrode 46. Control circuitry 38 typically comprises amplifiers and/orswitches, as are known in the art, which control either the amplitude orthe duty cycle, or both, of the voltage that is applied to eachelectrode 44. The pattern of amplitudes and/or duty cycles applied tothe electrodes determines the phase modulation profile of layer 48. Thecircuit components in circuitry 38 are typically fabricated as a siliconchip, which is then glued onto substrate 40. Alternatively, some or allof the components of circuitry 38 may be formed on a separate chip andconnected to substrate 40 by suitable bonding wires or otherconnections. In either case, the control circuitry can be located at theside of the array of electrodes, as shown in FIG. 2B, and there is noneed for any parts of the control circuitry to be located over theactive area of layer 48.

Circuitry 38 is able to modify the control voltages applied to each of aset of the excitation electrodes 44 (which may include all of theelectrodes) concurrently and independently. For example, circuitry 38may update the control voltages applied to all the odd electrodes in thearray alternately with all the even electrodes. This sort of approachscales readily to large electrode counts, and can thus be used to createelectrically-tunable optical systems with high pixel counts and fineresolution.

Electrodes 44 create an array of pixels whose pitch is defined by thecenter-to-center distance p between the electrodes, as shown in FIG. 2B.The width w of the conductive electrodes themselves defines the size ofthe pixels, while the gap g between the electrodes influences theseparation between the pixels and affects inter-pixel phase deviations.These dimensional parameters of the electrode array are compared to thelayer thickness d of electro-optical layer 48, shown in FIG. 2A.

In contrast to most liquid-crystal devices that are known in the art,the inter-electrode distance p of lens 34 is less than four times thethickness d of layer 48, and may be less than twice the thickness.Additionally or alternatively, the gap g between the electrodes may beless than the thickness of layer 48 or possibly even less than half thisthickness. In some implementations, even the pitch p may be less than d.This choice of dimensions permits a high effective fill factor ofpixels. Furthermore, the relatively thick layer 48 enables lens 34 togenerate a large range of different phase shifts, while the smallinter-electrode distance supports modulation of the refractive index,and hence the phase shift, with high resolution. The crosstalk betweenadjacent pixels that results from this choice of dimensions is actuallybeneficial in smoothing the phase modulation profile of the device, andthus more closely approximating the quadratic profile of a conventionallens. These features are analyzed further hereinbelow.

FIGS. 3A-3C schematically illustrate compound lens 26, which contains astatic component 50 and an electrically-tunable component, in the formof lens 34, in accordance with an embodiment of the present invention.FIG. 3A is a frontal view of lens 26, while FIGS. 3B and 3C areschematic sectional views of static component 50 and of theelectrically-tunable component, respectively, taken along the lineIII-III that is shown in FIG. 3A.

Static component 50 is configured as a lens, comprising a transparentmaterial having a curved exterior surface with a specified refractivepower. The interior surface of component 50 contains at least twooverlapping indentations 56 and 58, of respective depths h1 and h2, aswell as a groove 59, in order to accommodate the elements of lens 34.Specifically, the sizes and shapes of indentations 56 and 58 are chosenso that substrates 42 and 40 fit respectively into these indentations.Depth h1 corresponds to the combined thickness of substrate 42 and layer48, which is enclosed at its sides by sealing 54. Depth h2 is sufficientto contain substrate 40, possibly with an overlying encapsulation layer(not shown). Electrical connections 52 of control circuitry 38 aredisposed at the edge of substrate 40 and fit into groove 59. A socket orconnecting pads inside groove 59 link connections 52 to a small flexibleprinted circuit board 53 or other link to external power andinput/output circuits (not shown). Alternatively, dynamic lens 34 mayhave connectors on more than one side, with corresponding grooves 59formed in both sides of component 50.

The cruciform shape of dynamic lens 34, as defined by substrates 40 and42, is convenient for mounting in static lens, but other shapes mayalternatively be used. For example, in some cases, one of the substratesmay be smaller in both dimensions than the other substrate. As anotherexample, at least one side of at least one of the substrates may berounded, possible fitting the rounded shape of the static lens.Additionally or alternatively, multiple dynamic lenses may be stackedand encapsulated within the same static lens, such as two cylindricaldynamic lenses with perpendicular cylinder axes.

Electrodes of Non-Uniform Width

FIGS. 4A-4C schematically illustrate an electrically-tunable lens 66, inaccordance with an embodiment of the present invention. FIGS. 4A and 4Bare schematic frontal views of electrodes 64 formed on opposing sides 60and 62 of the lens. FIG. 4C is a schematic frontal view of lens 66,showing a superposition of electrodes 64 on the opposing sides of thelens. The intersection of the horizontal electrodes on side 60 with thevertical electrodes on side 62 defines an array of pixels 68.

When small pixels of uniform size are used in a lens, for example forvision correction, the regular pitch of the pixels causes substantialdiffraction, which can create disturbing ghost images on the retina ofthe subject. Lens 66 overcomes this problem by using electrodes 64having different, respective widths in the transverse direction, i.e.,the direction perpendicular to their axes. The widths of the successiveelectrodes are randomized with a predefined standard deviation around agiven mean. The inventors have found that when the widths of theelectrodes differ from one another with a standard variation that is atleast 10% of the mean width taken over all the electrodes, diffractioneffects are suppressed sufficiently to make the ghost images nearlyimperceptible. When the standard deviation of the width is 20% orgreater, the ghost images disappear almost entirely. Although lens 66comprises two arrays of mutually-perpendicular electrodes 64, theprinciples of this embodiment (as well as the embodiments that follow)are equally applicable, mutatis mutandis, to electrically-tunable lensesthat comprise only a single electrode array.

FIGS. 5A-5D are plots that schematically show the point spread functions(PSF) of retinal images formed using lens 66 (as shown in FIGS. 4A-4C)for different levels of pixel size variation, in accordance with anembodiment of the present invention. To compute the PSF, the eye isassumed to be a perfect lens, with a focal length of 17 mm and a pupilsize of 4 mm, while lens 66 has an average pixel pitch of 10 μm and isdriven to operate as a Fresnel lens with a focal length of 1 m. Theoptical axis of the eye is assumed to be displaced by 5 mm relative tothe center of lens 66, which tends to exacerbate the diffractioneffects.

Each of FIGS. 5A-5D shows the amplitudes of the diffraction orders inthe scheme defined above for different standard deviations, a, expressedin microns, of the electrode widths. FIGS. 5A-5D are scaled to emphasizehigh diffraction orders, and the zero-order peak value, which is notseen in the figures, is 0.02. For σ=1 μm, meaning a standard deviationof 10%, as shown in FIG. 5A, the first diffraction orders are reduced toless than 0.3% of the zero order and are thus barely perceptible, if atall. For higher standard deviations, the first diffraction orders arereduced further still. The total scattered light is not reduced and mayeven increase slightly with increasing standard deviation, but thisscatter does not disturb the subject's vision in any way.

FIGS. 6A-6C schematically illustrate an electrically-tunable lens 76, inaccordance with an alternative embodiment of the present invention.FIGS. 6A and 6B are schematic frontal views of electrodes 74 formed onopposing sides 70 and 72 of the lens. FIG. 6C is a schematic frontalview of lens 76, showing a superposition of electrodes 74 on theopposing sides of the lens. The intersection of the horizontalelectrodes on side 70 with the vertical electrodes on side 72 defines anarray of pixels 78.

As can be seen in FIGS. 6A and 6B, the widths of electrodes 74 not onlydiffer from one another, but also vary along the axes of the electrodes.(The axes of electrodes 74 are taken to be the centroidal axes along thelong dimensions of the electrodes, which are parallel to within a smalldeviation.) Consequently both the sizes and shapes of pixels 78 varyover the area of lens 76, and diffraction effects are negligible.Although the size variations in FIG. 6C are substantial, for clarity ofillustration, in practical implementations the variations will typicallybe smaller to enable accurate generation of the desired phase modulationprofile. The width variations of electrodes 74 may alternatively besmooth, rather than periodic with random amplitude as shown in thepresent figures.

FIGS. 7A-7C schematically illustrate an electrically-tunable lens 77, inaccordance with an alternative embodiment of the present invention.FIGS. 7A and 7B are schematic frontal views of electrodes 75 formed onopposing sides 71 and 73 of the lens. FIG. 7C is a schematic frontalview of lens 77, showing a superposition of electrodes 75 on theopposing sides of the lens. The intersection of the horizontalelectrodes on side 71 with the vertical electrodes on side 73 defines anarray of pixels 79.

As can be seen in FIGS. 7A and 7B, the widths of electrodes 75 areroughly constant, but their center points are displaced transversely byan amount that varies along the respective axes of the electrodes, whichare again taken to be the centroidal axes along the long dimensions ofthe electrodes. In this example, the transverse displacement of all theelectrodes is the same for any given distance measured along the axesfrom the edge of side 71 or 73, so that constant electrode width ismaintained, but alternatively, the widths and displacements of theelectrodes may vary. The varying transverse displacement of theelectrodes introduces a corresponding variation in the pitch of pixels79 in the spatial Fourier transform of lens 77, thus mitigating thediffraction peaks and ghost images that would otherwise be produced.

Although FIGS. 6A-6C and 7A-7C show lenses 76 and 77 with arrays ofelectrodes 74 or 75 extending across both sides, the principles of theseembodiments may similarly be implemented in cylindrical lenses, in whichthe electrodes of non-uniform width and/or displacement extend over onlyone side of the electro-optical layers. A pair of such cylindricallenses, with the electrode axes of the two lenses perpendicular to oneanother, can be used to achieve substantially the same focusing effectas in lens 74 or 75.

To maintain smoothness in the area between electrodes, the waveformsthat are applied by control circuitry 38 to drive adjacent electrodesshould generally be of the same polarity. An exception to this ruleoccurs when there are abrupt transitions in the phase modulationprofile, such as in Fresnel lenses: In this case, the control circuitrymay apply the respective control voltage waveforms with oppositepolarities to pairs of mutually-adjacent excitation electrodes in thevicinity of the abrupt phase transitions, in order to achieve a steeperchange in the phase modulation profile.

Additionally or alternatively, abrupt transitions in the phasemodulation profile of the electro-optical layer can be sharpened bydriving the electrodes at either side of the transition with voltagesthat overshoot or undershoot the nominal voltage for the given phasemodulation amplitude. When wide electrodes are used, however, asdescribed above, the overshoot and undershoot over the correspondinglywide areas in the electro-optical layer may have a negative impact onthe focal quality of the lens. This problem can also be addressed byusing electrodes having different, respective widths, as illustrated inthe figures that follow. The features of these embodiments may becombined with those of the preceding embodiments, and may be implementedin devices that include either a single array of excitation electrodeson one side of the electro-optical layer or two, mutually-perpendiculararrays on opposing sides.

FIG. 8 is a schematic frontal view of electrodes 82, 84 in anelectrically-tunable lens 80, in accordance with such an embodiment ofthe present invention. The design of lens 80 incorporates narrowelectrodes 84 interspersed with main modulation electrodes 82.Typically, the width of electrodes 84 is no more than half the width ofelectrodes 82, and may be still less. In fact, the width of electrodes84 may be no greater than four times the widths of the gaps between theelectrodes, or possibly even equal to or less than the gap width. Thiselectrode width may be less than the thickness of the electro-opticallayer itself.

The control voltage waveforms applied by the control circuitry areadjusted so that the abrupt transitions in the phase modulation profilewill occur in the vicinity of narrow electrodes 84. The narrowelectrodes are driven with overshoot or undershoot voltages, asappropriate, in order to sharpen these transitions. Because of thenarrow stripe width, however, the overshoot or undershoot is limited toa narrow spatial band at the phase transition border, while theremainder of the profile that is generated by main electrodes 82 remainssmooth and well formed.

In some embodiments of the present invention, when a phase modulationfunction with abrupt phase transitions, such as a Fresnel lens, isimplemented in a structure such as that shown in FIG. 8, the range ofphase modulation values supported by the electro-optical layer isincreased in proportion to a relation between the density of narrowelectrodes 84 relative to the spacing between the abrupt transitions inthe phase modulation function. A high density of narrow electrodesresults in a smaller increase of the required range of phase modulationvalues. For example, a Fresnel prism, with abrupt phase transitions of2π that are spaced a distance X apart, can be implemented using anelectro-optical layer supporting phase modulation values of 0 to 2π aslong as the electrode structure supports abrupt transitions from 2π to 0that are spaced X apart. In practical cases, however, in which theabrupt transitions are limited to the locations of narrow electrodes 84,and there is not a complete overlap between the locations of the narrowelectrodes and the abrupt transitions of the phase modulation function,the electro-optical layer should support a modulation range greater than2π. For instance, if the distance between narrow electrodes is 0.3X,some of the abrupt transitions will have to be spaced 1.2X apart inorder to match the narrow electrode locations (and in other places 0.9Xapart). In such cases, the phase modulation range of the electro-opticallayer should be 20% larger, covering the range of 0 to 2.4π. If thenarrow electrodes are spaced 2X apart, abrupt transitions will also be2X apart, and the active layer should then support a phase modulationrange of 0 to 4π.

FIG. 9 is a schematic frontal view of electrodes 82, 84 in anelectrically-tunable lens 86, in accordance with an alternativeembodiment of the present invention. This embodiment is similar to thatof FIG. 8, except that in lens 86, narrow electrodes 84 are paired inorder to enable more precise local control of the applied waveforms andphase modulation profile. In this case, one of the narrow electrodes inthe pair may be driven with an overvoltage, while the other receives anundervoltage.

FIG. 10 is a plot that schematically shows variation in phase modulationin two electrically-tunable lenses, in accordance with an embodiment ofthe present invention. A solid curve 90 shows a transition of 2π in thephase modulation profile of a lens with electrodes of uniform width,while a dashed curve 92 shows the sharper transition that is achievedusing narrow electrodes 84, as in the device of FIG. 8. The gap betweenthe electrodes in both cases is g=2 μm, and electrodes 84 are 2 μm wide.The electrodes are driven with a pulse-width modulated (PWM) signal ofamplitude 3.3 V. The low-phase electrodes, on the left side of FIG. 10,have a 5% duty cycle (V_(rms)=0.74 V), while the high-phase electrodes,on the right side, have a 20% duty cycle (V_(rms)=1.47 V). To generatecurve 92, electrode 84 at the high side of the transition is driven withan overshoot voltage, with 26% duty cycle (V_(rms)=1.68 V). The 2πtransition in the phase modulation function is visibly steeper whenusing narrow electrode 84 with voltage overshoot in this manner.

Avoiding Undesired Phase Variations

Referring back to FIGS. 2A and 2B, the most fundamental parameter of apixelated electrically-tunable lens is the pitch p, which determines thespatial sampling rate of the modulation function. The pitch should besmall enough to ensure that the phase modulation function does notchange significantly between pixels.

The gap width g between the electrodes is also important for achievingsmooth, continuous modulation functions, and should not be much largerthan the thickness d of the electro-optical layer. If g>>d, theelectrical field applied to the electro-optical material under the gapbetween the electrodes is significantly smaller than that below theelectrodes, resulting in significant discontinuities in the lightmodulation function. It is therefore desirable that the gap width g beless than the layer thickness d of the electro-optical layer, and evenless than half the layer thickness if possible.

FIG. 11 is a plot that schematically shows variations in phasemodulation between adjacent electrodes of an electrically-tunable lensfor different gap widths between the electrodes, in accordance with anembodiment of the present invention. The required phase modulation isconstant at 4.7 radians, and therefore the same voltage is applied tothe two electrodes on either side of the gap. The thickness of theliquid crystal layer is taken to be d=5 μm. Curves 93, 94 and 95 showthe phase change within the liquid crystal adjacent to the gap betweenthe electrodes for gap widths g=10 μm, g=5 μm, and g=2 μm, respectively.

Curves 93, 94 and 95 clearly illustrate the advantage of having a smallgap width when a smooth, continuous phase modulation function isrequired. In curve 93, with gap width larger than the electro-opticallayer thickness, the phase modulation drops from the required 4.7 toonly 0.9 radians between the electrodes. In curve 94, with the gap widthequal to the layer thickness, the drop in phase modulation is lessextreme, whereas in curve 95, for which the gap width is less than halfthe layer thickness, the phase modulation drops only to 4.5 radians.Thus, it is beneficial in most cases to use the smallest gap achievableunder the given manufacturing limitations (such as the feature size ofthe photolithography process).

FIG. 12 is a schematic sectional view of an electrically-tunable lens 96comprising a buffer layer 98 between electrodes 44 and electro-opticallayer 48 in the lens, in accordance with an alternative embodiment ofthe present invention. Buffer layer 98 comprises a transparentdielectric material, such as glass or a suitable polymer, typically atleast 0.2 μm thick, and in most cases at least 0.5 μm thick, whichseparates the excitation electrodes from the electro-optical layer. Theinterior surface of buffer layer 98 is adjacent to one side ofelectro-optical layer 48, while electrodes 44 on substrate 40 extendacross the exterior surface of the buffer layer. In the picturedembodiment, common electrode 46 on substrate 42 extends across theactive area of electro-optical layer 48. If an electrode pattern existson substrate 42 as well, however, an additional buffer layer (not shown)can be positioned on top of the bottom substrate, similar to layer 98.

The areas where the electric field in electro-optical layer 48 is mostaffected by the gaps between electrodes 44, resulting in the undesiredmodulation variations shown in FIG. 11, are adjacent to the gaps, nearsubstrate 40. Buffer layer 98 distances the electro-optical material inlayer 48 from these areas and for this purpose should have a thicknessthat is at least one-fourth of the gap width. Therefore, theelectro-optical material experiences a smoother electrical field, andthe phase modulation is smoother.

FIG. 13 is a plot that schematically shows variation in phase modulationbetween pixels in lens 96 and in a comparable lens without a bufferlayer, in accordance with an embodiment of the present invention. Inthis example, the thickness of electro-optical layer 48 is d=5 pin,while electrodes 44 have width w=9 μm and are separated by gaps of widthg=3 μm. A curve 100 shows the phase variations in layer 48 without abuffer layer, while a curve 102 shows the phase variations with theaddition of a buffer layer 98 of thickness b=1 μm. The buffer layersmooths the light modulation function and achieves more continuousmodulation.

On the other hand, the buffer layer will also smooth the 2nπ abruptphase transitions in Fresnel-type profiles, thus increasing themodulation errors in these areas. These errors can be reduced bypositioning additional narrow electrodes on the side of buffer layer 98that is directly adjacent to electro-optical layer 48 (the lower side inFIG. 12). These latter electrodes may be used in implementing steepmodulation changes in the manner described above with reference to FIGS.7-9.

Driving Waveforms for Separable Modulation Functions

FIGS. 14A-14C schematically illustrate an electrically-tunable lens 112,which is used in generating phase modulation profiles that are separablein the X- and Y-directions, in accordance with an embodiment of thepresent invention. FIGS. 14A and 14B are schematic frontal views ofelectrodes 110 formed on opposing sides 106 and 108, respectively, ofthe electro-optical medium in lens 112 (not shown in these figures).FIG. 14C is a schematic frontal view of lens 112, showing a matrix ofpixels 114 defined by the superposition of electrodes 110 on theopposing sides of the device.

As noted earlier, lens 112 can be driven to implement phase modulationprofiles e^(iø(x,y)) that are separable, meaning that they can bedecomposed into a product of two one-dimensional functions,e^(iø(x,y))=e^(iø) ^(x) ^((x))·e^(iø) ^(y) ^((y)). In other words, theseparable phase modulation function is decomposed into twoone-dimensional phase modulation functions: φ(x,y)=φ_(X)(x)+φ_(Y)(y).The one-dimensional functions can be quantized to N quantization levelsof the phase angle θ, for example

${\theta_{k} = {\frac{k}{N}\theta_{\max}}},{k = {{0\ldots \; N} - 1.}}$

thus, in some embodiments of the present invention, respective voltagewaveforms V_(x,k)(t) and V_(y,k)(t) for the X- and Y-axis electrodes110, on sides 106 and 108 of lens 112, are defined corresponding tothese quantization levels, such that for every pair of levels (k₁,k₂),applying V_(x,k) ₁ (t) to a vertical electrode and V_(y,k) ₂ (t) to ahorizontal electrode will result in a phase modulation of θ=θ₀+θ_(k) ₁+θ_(k) ₂ in the pixel 114 at the intersection of the electrodes. In thisexpression, θ₀ is a constant phase common to all pixels, which does notaffect the light propagation through lens 112.

For example, Table I below shows the phase modulation of a pixel as afunction of the applied voltage waveforms on the vertical and horizontalelectrodes defining the pixel, assuming quantization to four phasemodulation levels,

${\theta_{k} = \frac{k\; \pi}{2}},{k = {{0\ldots \; 3}:}}$

TABLE I Y X V_(y,0)(t) V_(y,1)(t) V_(y,2)(t) V_(y,3)(t) V_(x,0)(t) θ₀θ₀ + π/2 θ₀ + π θ₀ + 3π/2 V_(x,1)(t) θ₀ + π/2 θ₀ + π θ₀ + 3π/2 θ₀ + 2πV_(x,2)(t) θ₀ + π θ₀ + 3π/2 θ₀ + 2π θ₀ + 5π/2 V_(x,3)(t) θ₀ + 3π/2 θ₀ +2π θ₀ + 5π/2 θ₀ + 3πIn another embodiment of the present invention, the voltage waveformsV_(x,k)(t) and V_(y,k)(t) are defined in a way that enables moreefficient modulation by reducing the dynamic range of the phasemodulation. Specifically, the voltage waveforms V_(x,k)(t) andV_(y,k)(t) are defined such that for any (k₁,k₂), applying V_(x,k) ₁ (t)on a vertical electrode and V_(y,k) ₂ (t) on a horizontal electrode willresult in a phase modulation of θ_(k) ₁ _(,k) ₂ =θ₀+θ_(k) ₁ +θ_(k) ₂−2πl_(k) ₁ _(,k) ₂ , wherein l_(k) ₁ _(,k) ₂ is a non-negative integer.A simple example of this sort of function is modular phase addition, asgiven by θ_(k) ₁ _(,k) ₂ =(θ₀+θ_(k) ₁ +θ_(k) ₂ ) mod 2nπ. As notedearlier, the value of the integer n need not be constant and may varyfor different pairs of component waveforms.

As a result of this approach, thinner electro-optical layers can beused, enabling steeper phase transitions when abrupt changes arerequired. Alternatively, this sort of approach can be used to decreasethe number of phase discontinuities in the modulation function.

Table II illustrates this approach for four quantization levels (as inTable I), with n=1, i.e., θ_(k) ₁ _(,k) ₂ =(θ₀+θ_(k) ₁ +θ_(k) ₂ ) mod2π:

TABLE II Y X V_(y,0)(t) V_(y,1)(t) V_(y,2)(t) V_(y,3)(t) V_(x,0)(t) θ₀θ₀ + π/2 θ₀ + π θ₀ + 3π/2 V_(x,1)(t) θ₀ + π/2 θ₀ + π θ₀ + 3π/2 θ₀V_(x,2)(t) θ₀ + π θ₀ + 3π/2 θ₀ θ₀ + π/2 V_(x,3)(t) θ₀ + 3π/2 θ₀ θ₀ + π/2θ₀ + πWhen the voltage waveforms are defined according to Table II, theelectro-optical layer is required to support a dynamic phase modulationrange of 3π/2, as compared to 3π in the example of Table I, thusenabling a thinner active layer and steeper transitions when phasediscontinuities are required.

Alternatively, for a given electro-optical layer thickness, the aboveapproach can be used to reduce the number of phase discontinuities. Inthis case, the driving scheme is similar to that shown in Table II, buteach axis is modulated by values ranging from 0 to 4π (instead of the 2πlimit of Table II), and the phase summation over the vertical andhorizontal electrodes is modulo 4π (rather than 2π in Table II). Whenusing the scheme of Table I, a phase discontinuity occurs every time therequired modulation crosses 2π, either in the X- or in the Y-axis, evenif the electro-optical layer supports a dynamic modulation range of 4πor more. On the other hand, when a scheme similar to that shown in TableII is used, but with n=2 (modulo 4π), a phase discontinuity occurs everytime the required modulation crosses 4π, with cylindrical symmetry.

FIG. 15 is a schematic representation of a phase modulation profile 116generated by lens 112 when driven as described above, in accordance withan embodiment of the present invention. Profile 116 emulates a Fresnellens, with an amplitude scale 118 of phase shift ranging from 0 to 4π.Profile 116 includes a central lobe 120 and peripheral orders 122,separated by abrupt phase transitions of amplitude 4π. This schemesignificantly decreases the number of phase discontinuities, relative toa Fresnel profile with transitions at every 2π change of amplitude, andaccordingly reduces the areas that these abrupt transitions occupywithin the phase modulator, thus improving the accuracy of the phasemodulation function.

FIG. 16 is a plot that schematically illustrates a phase modulationcurve as a function of the voltage applied across electrodes 110 in lens112, in accordance with an embodiment of the present invention. As shownin FIG. 16, phase modulations of θ₀, θ₀+π/2, θ₀+π and θ₀+3π/2 areobtained by applying root mean square (RMS) voltages of 1.02 V, 1.14 V,1.29 V and 1.46 V, respectively, across the pixels.

FIGS. 17A-17D and 18A-18D are plots that schematically illustratecomponent voltage waveforms applied to electrodes 110 in a duty cyclemodulation scheme of this sort, in accordance with an embodiment of thepresent invention. FIGS. 17A-17D show four different waveforms, withdifferent duty cycles, that are applied to the X-electrodes, on side 106of lens 112, while FIGS. 18A-18D show the voltage waveforms, likewisewith differing duty cycles, that are applied to the Y-electrodes, onside 108. The voltage waveforms in this example are limited to threevalues: 0 and ±2.5 V, and provide phase modulations of θ₀, θ₀+π/2, θ₀+πand θ₀+3π/2 as required for the modulation scheme of Table II.Alternatively, other sets of waveforms and modulation schemes can beconstructed, with the same or larger numbers of quantization levels. Forexample, analog driving can be used, in which each electrode can receiveone of four or more voltage values in each time slot.

In the example shown in FIGS. 17A-17D and 18A-18D, the voltage appliedon the electro-optical layer is of alternating polarity (AC), at afrequency larger than the response time of the electro-optical media(for example, larger than 100 Hz for liquid crystal media). Thewaveforms shown in FIGS. 17A-17D and 18A-18D are of half a cycle. Inpractice, these waveforms are duplicated with alternating polarity at alarge enough rate so that the response of the electro-optical layer willdepend only on the RMS value of the applied voltage.

FIGS. 19A-19G are plots that schematically illustrate voltage waveformsgenerated across the electro-optical layer in lens 112 as a result ofapplying different combinations of the waveforms of FIGS. 17A-17D andFIGS. 18A-18D to electrodes 110 on opposing sides 106 and 108 of thelens. For the sake of brevity and simplicity, only a subset of thesixteen combined waveforms is shown. The titles of the plots includesthe RMS voltage values of the waveforms, which match the RMS valuesshown in FIG. 16 and correspond to the phase shift values in Table II.

The voltage differences between the X- and Y-electrodes in this exampleare limited to the values 0, ±2.5 V, and ±5 V. Using a larger number ofvoltage values can add more degrees of freedom to the waveforms, whichcan be used to eliminate the short pulses that appear in some of thewaveforms.

Electrically-Tunable Cylindrical Lenses in Series

FIG. 20 is a schematic side view of an optical system 130, in accordancewith another embodiment of the invention. As described earlier, system130 comprises two electrically-tunable cylindrical lenses 132 and 134arranged in series, with mutually-perpendicular cylinder axes. (Forexample, the cylinder axis of lens 132 may be vertical in the plane ofthe figure sheet, while that of lens 134 points into the sheet.) In thismanner, lenses 132 and 134 can be controlled together in order toemulate a spherical or aspheric lens with a two-dimensional modulationprofile, as the superposition of the respective modulation profiles oflenses 132 and 134.

For ease of manufacturing, it is convenient that lenses 132 and 134 havethe same structure, for example as shown in FIGS. 2A and 2B. Assumingelectro-optical layer 48 to comprise a polarization-dependent medium,such as a polarization-dependent liquid crystal, the polarization axesof lenses 132 and 134 will then be mutually-perpendicular, as well.Consequently, in the absence of correction, lenses 132 and 134 willoperate on different polarizations (and thus will be ineffective as atwo-dimensional lens). In order to overcome this limitation, system 130comprises a polarization rotator 136, such as a quarter-wave plate.Thus, assuming the light entering system 130 to be vertically polarized,in alignment with the polarization axis of lens 132, rotator 136 willrotate the polarization axis by 90° so that the axis is aligned with thepolarization axis of lens 134 when the light is incident on lens 134.

Although the above description and the corresponding figures relateseparately, for the sake of clarity, to various different features ofelectrically-tunable lenses, these features should by no means beconsidered to be mutually exclusive. On the contrary, those skilled inthe art will understand, after reading the above description, that thesefeatures may be combined in order to achieve still further enhancementof device performance. It will thus be appreciated that the embodimentsdescribed above are cited by way of example, and that the presentinvention is not limited to what has been particularly shown anddescribed hereinabove. Rather, the scope of the present inventionincludes both combinations and subcombinations of the various featuresdescribed hereinabove, as well as variations and modifications thereofwhich would occur to persons skilled in the art upon reading theforegoing description and which are not disclosed in the prior art.

1. An optical device, comprising: an electro-optical layer, having aneffective local index of refraction at any given location within anactive area of the electro-optical layer that is determined by a voltagewaveform applied across the electro-optical layer at the location;conductive electrodes extending over opposing first and second sides ofthe electro-optical layer, the electrodes comprising an array ofexcitation electrodes, which extend along respective, mutually-parallelaxes in a predefined direction across the first side of theelectro-optical layer, and which comprises at least first and secondelectrodes having different, respective widths in a transversedirection, perpendicular to the axes; and control circuitry, which iscoupled to apply respective control voltage waveforms to the excitationelectrodes and to modify the control voltages applied to each of theexcitation electrodes concurrently and independently so as to generate aspecified phase modulation profile in the electro-optical layer.
 2. Thedevice according to claim 1, wherein the respective widths of theelectrodes differ from one another with a standard variation that is atleast 10% of a mean width of all the electrodes.
 3. The device accordingto claim 1, wherein the respective widths of at least some of theelectrodes vary along the respective axes of the electrodes.
 4. Thedevice according to claim 1, wherein the array of excitation electrodescomprises a first array of first excitation electrodes, extending in afirst direction across the first side of the electro-optical layer, andwherein the conductive electrodes comprises a second array of secondexcitation electrodes, which extend in a second direction, perpendicularto the first direction, across the second side of the electro-opticallayer, and which comprises at least third and fourth electrodes havingdifferent, respective widths.
 5. The device according to claim 1,wherein the conductive electrodes comprise a common electrode,positioned over the active area on the second side of theelectro-optical layer.
 6. Apparatus comprising first and second opticaldevices according to claim 5, wherein the first and second opticaldevices are arranged in series, and wherein the excitation electrodes inthe second optical device are oriented in a direction orthogonal to theexcitation electrodes in the first optical device.
 7. The apparatusaccording to claim 6, wherein the first and second optical devicescomprise respective, first and second electro-optical layers that arepolarization-dependent and are arranged such that the first opticaldevice modulates light in a first polarization, while the second opticaldevice modulates the light in a second polarization, different from thefirst polarization, and wherein the apparatus comprises a polarizationrotator positioned between the first and second optical devices so as torotate the light from the first polarization to the second polarization.8. The device according to claim 1, wherein the first and secondelectrodes have respective first and second widths, such that the firstwidth is at least twice the second width, and wherein the controlcircuitry is configured to apply the respective control voltagewaveforms so that the specified phase modulation profile has an abrupttransition that occurs in a vicinity of at least one of the secondelectrodes.
 9. The device according to claim 8, wherein generation ofthe specified phase modulation profile causes the device to function asa Fresnel lens.
 10. The device according to claim 8, wherein theelectrodes comprise parallel stripes of a transparent conductivematerial having gaps between the stripes of a predefined gap width, andwherein the second width of the second electrodes is no greater thanfour times the gap width.
 11. The device according to claim 8, whereinthe second width of the second electrodes is less than a layer thicknessof the electro-optical layer.
 12. The device according to claim 8,wherein the phase modulation profile has multiple abrupt transitionsthat occur in respective vicinities of corresponding ones of the secondelectrodes, and wherein the electro-optical layer is configured toprovide a range of phase modulation values that is proportional to arelation between a density of the second electrodes relative to aspacing between the abrupt transitions in the phase modulation function.13. The device according to claim 1, wherein the electro-optical layercomprises a liquid crystal.
 14. An optical device, comprising: anelectro-optical layer, having an effective local index of refraction atany given location within an active area of the electro-optical layerthat is determined by a voltage waveform applied across theelectro-optical layer at the location, the electro-optical layer havingopposing first and second sides and a layer thickness equal to adistance between the first and second sides; conductive electrodesextending over the first and second sides of the electro-optical layer,the electrodes comprising an array of excitation electrodes comprisingparallel stripes of a transparent conductive material having gapsbetween the stripes of a gap width that is no greater than 2 μm and isless than the layer thickness of the electro-optical layer; and controlcircuitry, which is coupled to apply respective control voltagewaveforms to the excitation electrodes so as to generate a specifiedphase modulation profile in the electro-optical layer.
 15. The deviceaccording to claim 14, wherein the gap width is less than half the layerthickness.
 16. The device according to claim 14, wherein theelectro-optical layer comprises a liquid crystal. 17-30. (canceled) 31.A method for producing an optical device, the method comprising:providing an electro-optical layer, having an effective local index ofrefraction at any given location within an active area of theelectro-optical layer that is determined by a voltage waveform appliedacross the electro-optical layer at the location; positioning conductiveelectrodes over opposing first and second sides of the electro-opticallayer, the electrodes comprising an array of excitation electrodes,which extend along respective, mutually-parallel axes in a predefineddirection across the first side of the electro-optical layer, and whichcomprises at least first and second electrodes having different,respective widths in a transverse direction, perpendicular to the axes;and coupling control circuitry to apply respective control voltagewaveforms to the excitation electrodes and to modify the controlvoltages applied to each of the excitation electrodes concurrently andindependently so as to generate a specified phase modulation profile inthe electro-optical layer.
 32. The method according to claim 31, whereinthe respective widths of the electrodes differ from one another with astandard variation that is at least 10% of a mean width of all theelectrodes.
 33. The method according to claim 31, wherein the respectivewidths of at least some of the electrodes vary along the respective axesof the electrodes.
 34. The method according to claim 31, wherein thearray of excitation electrodes comprises a first array of firstexcitation electrodes, extending in a first direction across the firstside of the electro-optical layer, and wherein positioning theconductive electrodes comprises positioning a second array of secondexcitation electrodes to extend in a second direction, perpendicular tothe first direction, across the second side of the electro-opticallayer, and wherein the second array comprises at least third and fourthelectrodes having different, respective widths. 35-74. (canceled)